Now that the new academic year is about to start, I thought it would be a good idea to mention a good way to keep track of research ideas in graduate school and beyond. Whether you are a graduate student searching for a dissertation problem, a postdoc or a faculty member engaged in research, you must be ready when an idea strikes you. Research questions and ideas do not usually just fall on our lap; you have to look for them, remember them and look for connections among them. A good way to organize these research ideas is to keep a journal (a paper or electronic notebook) where you write them down. Here are some suggestions.
Get a notebook that you use only for the purpose of writing mathematical ideas that you might pursue in your research and keep the notebook handy all the time. Bring the notebook to seminars, colloquia, and conference talks. Those are typical places where something that the speaker says can trigger an idea. The goal is to write down the essence of the idea and you can give it more thoughtful consideration later.
Write down open questions, problems, ideas of solution methods, references, and conjectures. You need just enough information to remember the context of the research idea and where to look to get started reading about it.
You are likely to collect many more research ideas than you can work on so do not expect to be able to work on all of the problems you write down. However, it is good to read your notebook periodically in order to refresh your memory and look for connections between the entries. Sometimes two ideas from different years can be related and produce a theme. These themes are excellent material for a possible grant proposal or future project.
You might schedule some time every couple of weeks to look at your notebook and spend a little time following up on something you wrote. You might find that it is part of a larger problem or that it is a piece of a larger question.